204 XIII INTEGRATION that for h to belong to /", it is necessary and sufficient that for each a E R the charac- teristic function of the set h'1(]oct + oo[) should belong to /. (c) Show that if k e / and /E &l, then U • (hf) = h(U •/) almost everywhere. (Reduce to the case where h = 9?A > and show that the integral of (1 — <p&)(U • (<PA/)) isO.) (d) Show that the set i^ of functions of the form h$+g~-U-g, where h e / and # e J2?i , is dense in ^ . (Use Section 12.15, Problem 4(f ) ; show that ifue&£ is such that u(g — U • g) dp, = 0 for all # e «^R , then w necessarily belongs to /, and deduce that w2<D ^ = 0.) (e) If /= h^+g — U-gei^, then the sequence (Rw(/, <D)) converges to h almost everywhere, and we have Ni(/z<D) <J NI(/). (For the latter assertion, consider the function u = sgn h and evaluate ufd\L, bearing in mind that u E /.) Deduce that there exists a continuous endomorphism jR<i> of Li , of norm :g 1, whose image is the closure in L£ of the vector subspace formed by the classes #<D where he /, and such that R* -/is equal to WS> if /== hQ + g-U-ge-T. (f) For each function /e J&?£, we denote by ^ -/any function in the class #<> •/ Deduce from (e) that the sequence (Rn(/5 <£)) converges to 3>~l(R<s> •/) almost every- where. Furthermore, for each aeR, the set of points xeX such that (R® -/)(X)^ a^C*) is such that its characteristic function belongs to ,/, and for each function he/ we have | h(R<t> •/) d/z = f hfd^. (If/i e ^ is such that NI(/—/I) ^ e, show that the set of XEX such that R*(/-/i, <&)(*) ^£1/2, or ^(/-/i)W > £1/2$>W, has measure ^ 2e1/2 with respect to <D • p,, by using Section 13.11, Problem 18(c).) (g) Deduce that for any two functions /, g e J§?R such that g > 0, the sequence (Rn(/> #)) converges almost everywhere to a finite limit in the set B of points x e X such that (Un • g)(x) > 0 for at least one value of the integer n ]> 0 (Chacon-Ornstein theorem). 6. (a) Let o^ < u2 be two real numbers, let B be the vertical strip ol <j &s ^ a2 in C, and let /be a function holomorphic in £ and continuous in B. Suppose also that 0) I /CO I SI M for s = cr i + it and ^ = a2 + it, where ^ e R is arbitrary; (ii) there exist two constants a > 0, A > 0 such that \f(a + it) \ g A • efl|t| for all cr+ r/eB. Show that under these assumptions we have |/(s)| ^ M throughout B. (Use the Phragmen-Lindelof principle as in Section 9.5, Problem 17, by taking g(s) = e*2.) (b) If /is continuous and bounded in B and holomorphic in S, and if we put L(cr) = sup |/(<j H- fr)l f°r ^1 ° e fri* ^i], then we have (three-line theorem). (Proceed as in Section 9.5, Problem 10, by considering the function s >-> eas/(.s) for a suitable «eR. Hadamard's three-circle theorem is a par- ticular case of (*).) 7. Let X, Y be two locally compact spaces and let p, (resp. v} be a positive measure on X (resp. Y). Let E <=• L£(X, p.) be the space of classes of ju,-integrable step functions, and let U be a linear mapping of E into the space Lfoc.cfY, v) (Section 13.13). For each pair of numbers p, q (finite or not) belonging to the interval [1 , + oo ] of R, thef% such that '£/ • h = h almost everywhere